確率統計 特論 (Probability and Statistics)

Transcription

1 確率統計 特論 (Probability & Statistics) Oct. 4, 2017 共通基礎科目 確率統計 特論 (Probability and Statistics) Todays topics what is probability? probability space 来嶋秀治 (Shuji Kijima) システム情報科学研究院情報学部門 Dept. Informatics, ISEE

2 Preview About this class

3 Evaluation exercises ( レポート課題 ) =: z (/10) Almost every class (10 回程度 ) Submit hard copy at the end of class, or to kijima@inf.kyushu-u.ac.jp before then. midterm ( 中間試験 ): (early) Nov., =:x (/50) final ( 期末試験 ): late Jan or early Feb, =:y (/50) score = x+y (+ z), something like that supplemental option for credit not mandatory, but recommend. passing mark = 60. No makeup exam ( 追試は無い ) 3

4 Lecture Use slides and/or board. as much as possible sorry. Language: speak in Japanese, write in English. mathematical backgrounds (undergraduate level) e.g., calculus ( 微積分 ), linear algebra ( 線形代数 ), etc Questions are welcome in English or Japanese. Do not hesitate! Check the web page! you can find exercise, keywords, and schedule. Office hour: Fri( 金曜日 ) 16:40 18:10 (W2, 7F, 741) Topics see 4

5 References 5 藤澤洋徳, 確率と統計, 現代基礎数学 13, 朝倉書店, 2006, 4000 円. M. Mitzenmacher, E. Upfel, Probability and Computing: Randomized Algorithms and Probabilistic Analysis, Cambridge University Press, ( 邦訳 ) 小柴健史, 河内亮周, 確率と計算 -- 乱択アルゴリズムと確率的解析 --, 共立出版, [ 確率論 ] 伏見正則, 確率と確率過程, 朝倉書店, 2002, 2800 円 [ 統計学 ] 日本統計学会, 統計学 ( 統計検定 1 級対応 ), 東京図書, 東大教養学部統計学教室編, 統計学入門, 東京大学出版会, 1991, 2940 円. Check the web page!

6 Requirement ( 前提知識 ) 6 Analysis ( 解析学 ) Differentiation ( 微分 ) d dx log x? Integration by parts ( 部分積分 ) b a x exp( x) dx Linear algebra ( 線形代数学 ) Bases ( 基底 ) Eigenvalues/diagonalization ( 固有空間 / 対角化 )

7 What is Probability? Examples

8 Ex 1. Monty Hall problem --- ask Marilyn 8 You are given the choice of three doors: Behind on door is a car; behind the others goats. You pick a door, say A. The host (Monty), who knows what's behind the doors, opens another door, say C, which he knows has a goat. He then says to you, Do you want to pick door B? Question Is it to your advantage to switch your choice? 図 : wikipedia モンティーホール問題 より

9 Ex 1. A modified Monty Hall problem --- ask Marilyn 9 You are given the choice of 26 doors: Behind on door is a car; behind the others goats. You pick a door, say A. The host (Monty), who knows what's behind the doors, opens all other doors, say C to Z, except for B, each of which he knows has a goat. He then says to you, "Do you want to pick door B? Question Is it to your advantage to switch your choice? 図 : wikipedia モンティーホール問題 より

10 Ex 2. Bertrand paradox 10 Consider an equilateral triangle inscribed in a circle. Suppose a chord of the circle is chosen at random. Question What is the probability that the chord is longer than a side of the triangle?

11 Ex 2. Bertrand paradox 11 Consider an equilateral triangle inscribed in a circle. Suppose a chord of the circle is chosen at random. Question What is the probability that the chord is longer than a side of the triangle (=:x)? Answer 1: The "random midpoint" method. Choose a point anywhere within the circle, and construct a chord with the chosen point as its midpoint. The chord is longer than x iff the chosen point within a small circle. the probability is 1/4

12 Ex 2. Bertrand paradox 12 Consider an equilateral triangle inscribed in a circle. Suppose a chord of the circle is chosen at random. Question What is the probability that the chord is longer than a side of the triangle (=:x)? Answer 2: The "random radius" method: Choose a radius of the circle and a point on the radius. the chord through this point and perpendicular to the radius. The chord is longer than x iff the chosen point on a blue line. the probability is 1/2

13 Ex 2. Bertrand paradox 13 Consider an equilateral triangle inscribed in a circle. Suppose a chord of the circle is chosen at random. Question What is the probability that the chord is longer than a side of the triangle (=:x)? Answer 3: The "random endpoints" method:: Choose two random points on the circumference of the circle and draw the chord joining them. The chord is longer than x iff /3 2 /3 the probability is 1/3

14 Ex 2. Bertrand paradox 14 Consider an equilateral triangle inscribed in a circle. Suppose a chord of the circle is chosen at random. Question What is the probability that the chord is longer than a side of the triangle (=:x)? What does a chord of the circle is chosen at random mean?

15 Probability Space Definitions Axiom Terminology

16 Definition: Probability Space 16 A probability space is defined by (Ω, F, P) Ω: sample space( 標本空間 ); a set of elementally events( 標本点 ), an event( 事象 ) is a subset of Ω. F: -algebra ( 2 Ω ); a set of events. P: probability measure( 確率測度 ); a function F R, probability of an event. F is a -algebra 1. F contains the empty set: F 2. F is closed under complements: A F ഥA F 3. F is closed under countable unions: A i F ڂ i A i F

17 Definition: Probability Space 17 A probability space is defined by (Ω, F, P) Ω: sample space( 標本空間 ); a set of elementally events( 標本点 ), an event( 事象 ) is a subset of Ω. F: -algebra ( 2 Ω ); a set of events. P: probability measure( 確率測度 ); a function F R, probability of an event. P is a probability measure (Kolmogorov axioms) 1. P A 0 for any A F. 2. P Ω = Any countable sequence of mutual exclusive events A 1, A 2, satisfies that P A 1 A 2 = P A 1 + P A 2 +.

18 Example: Probability Space 18 A probability space is defined by (Ω, F, P) Ω: sample space( 標本空間 ); a set of elementally events( 標本点 ), an event( 事象 ) is a subset of Ω. F: -algebra ( 2 Ω ); a set of events. P: probability measure( 確率測度 ); a function F R, probability of an event. Ex. 1. Die Ω = {1,2,3,4,5,6}, F = 2 Ω P A = A 6 for any A Ω.

19 Example: Probability Space 19 A probability space is defined by (Ω, F, P) Ω: sample space( 標本空間 ); a set of elementally events( 標本点 ), an event( 事象 ) is a subset of Ω. F: -algebra ( 2 Ω ); a set of events. P: probability measure( 確率測度 ); a function F R, probability of an event. Ex. 2. product of a pair of dice cast Suppose we can only observe the parity of the product x y of a pair of casts (x,y). ={(1,1), (1,2), (1,3),, (4,5), (4,6), (5,6)}, F={, even, odd, }, P( )=0, P(even)=?, P(odd)=?, P( )=1. => exercise

20 Property of Probability (1/2) 20 Kolmogolov s axiom (P1), (P2), (P3) implies the following. Thm. 1 Let Ω, F, P be an arbitrary probability space. (1) P = 0. (2) If A B (A, B F), then P B A = P B P A. (rem. B A = B ഥA ) (3) If A B (A, B F), then P A P B. (4) P A 1 for any A F. (5) P ഥA = 1 P(A) for any A F.

21 Property of Probability (2/2) 21 Kolmogolov s axiom (P1), (P2), (P3) implies the following. Thm. 1. (cont.) (6) P P A B = P A + P B P A B for any A F. (7) If A 1, A 2, F satisfy A 1 A 2, then P ڂ i A i = lim n P A n (8) If A 1, A 2, F satisfy A 1 A 2, then P ځ i A i = lim n P A n (9) For any A 1, A 2, F, P ڂ i A i = 0 i 0, P A i = 0